In practice, network operators tend to choose sparse communication topologies to cut costs, and the concurrent use of a communication network by multiple users commonly results in feedback delays. Our goal was to obtain the optimal sparse feedback control matrix $K$. For this, we proposed a sparse optimal control (SOC) problem governed by the cyber-physical system with varying delay, to minimize $||K|{|}_0$ subject to a maximum allowable compromise in system cost. A penalty method was utilized to transform the SOC problem into a form that was constrained solely by box constraints. A smoothing technique was used to approximate the nonsmooth element in the resulting problem, and an analysis of the errors introduced by this technique was subsequently conducted. The gradients of the objective function concerning the feedback control matrix were obtained by solving the state system and a variational system simultaneously forward in time. An optimization algorithm was devised to tackle the resulting problem, building on the piecewise quadratic approximation. Finally, we have presented of simulations.

This paper extends a result of isolated calmness for nuclear norm regularized convex optimization problems to Ky Fan $k$-norm regularized convex optimization problems. We find that there exists a certain equivalence relationship among the critical cones of the Ky Fan $k$-norm function and its conjugate as well as the "sigma term", namely, the conjugate function of the parabolic second-order directional derivative of the Ky Fan $k$-norm. By establishing the equivalence between the primal (dual) strict Robinson constraint qualification (SRCQ) and the dual (primal) second-order sufficient condition (SOSC), we derive a series of complete characterizations of the robust isolated calmness of the Karush-Kuhn-Tucker (KKT) mapping for Ky Fan $k$-norm regularized convex matrix optimization problems. The obtained results enrich the stability theory of the Ky Fan $k$-norm regularized convex optimization problems and further enhance the usability of the related algorithms.